A 10 kilogram object suspended from the end of a vertically hanging spring stret

Question

A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is distributed from its rest state by the force F(t) = 130 cos(10t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. 1. Determine the spring constant k. k = Newtons/meter 2. Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y, y', y", t. Differential equation: help (equations) Initial conditions: y(0) = and y'(0) = help (numbers) 3. Solve the initial value problem for y(t). y(t) = help (formulas) 4. Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 lessthanorequalto t

Explanation / Answer

Note that (y + 0.098) m is the total extension of the spring.

Let k be the spring constant; then k*(y + 0.098) Newtons is the force of the spring on the object in the upwards direction.

We now apply Newton’s law of motion ma=F in the downward direction.

We obtain 10y’’ = 10*9.8-k*(y + 0.098) + 130 cos 10t.

In equilibrium (y’'= y = 0 and no force), we have 0 = 98 - .098*k,

So spring constant k =1000 Newtons/meter

The equation now becomes y'' + 100y = 13cos 10t also y(0) = 0 and y’(0)=0

The characteristic equation of the complementary equation is r2+100=0,

so r = +10i, -10i

the complementary solution is Yc = C1 *cos 10t + C2*sin 10t

particular solution:

assume a solution of the form y = A *t * sin 10t+ B* t *cos 10t

thus putting assumed solution of y in our differential equation and comparing coefficients we will get A= 13/20 and B=0

Yp = 13/20 * t * sin 10t

over all solution to the differential equation

y(t) = Yc+Yp = C1*cos 10t + C2*sin 10t + 13/20 * t * sin 10t

again put y(0) =0

C1 = 0

also y’ (0) = 0

C2 = 0

thus solution to the differential equation after putting initial conditions : y(t) = 13/20 * t * sin 10t

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